eulogize 发表于 2025-3-27 00:46:08

Applications of Jacobi’s Elliptic FunctionsIn the previous chapter we defined Jacobi’s elliptic function sn as the inverse function of the incomplete elliptic integral of the first kind, introduced cn and dn and studied their properties. These Jacobi’s elliptic functions appear in various problems, from which we pick up two applications to physics in this chapter.

HACK 发表于 2025-3-27 02:24:56

Elliptic CurvesExercise 7.2 Prove the above proposition. (Hint: (i) follows from the fact that 𝜑(𝑧) does not have multiple roots. For (ii) and (iii) use local coordinates 𝑧 (around a point which is not a branch point) and 𝑤 (around branch points), after checking that they really are local coordinates.)

从属 发表于 2025-3-27 09:22:21

Complex Elliptic IntegralsThe expression (8.4) means that ‘in the homology group any closed curve is equivalent to a curve which goes around 𝐴 several times and then goes around 𝐵 several times’. This can be explained in the following way.

groggy 发表于 2025-3-27 10:03:58

http://reply.papertrans.cn/31/3078/307796/307796_34.png

神圣在玷污 发表于 2025-3-27 16:50:25

http://reply.papertrans.cn/31/3078/307796/307796_35.png

Hay-Fever 发表于 2025-3-27 18:24:40

The Weierstrass ℘-FunctionIn the previous chapter we defined elliptic functions as meromorphic functions on an elliptic curve = doubly periodic meromorphic functions on . and studied their properties. In particular, we gave several examples of elliptic functions which are obtained immediately from the definitions.

冒失 发表于 2025-3-27 23:45:22

http://reply.papertrans.cn/31/3078/307796/307796_37.png

neuron 发表于 2025-3-28 04:46:24

Characterisation by Addition FormulaeIn the previous chapter we proved that rational functions, rational functions of an exponential function and elliptic functions have addition theorems (algebraic addition formulae). Are there other functions which have algebraic addition formulae? The next Weierstrass–Phragmén theorem1 answers this question.

刀锋 发表于 2025-3-28 10:17:16

https://doi.org/10.1007/978-3-031-30265-7elliptic functions; elliptic integrals; complex analysis; application to physics; Riemann surfaces; ellip

Neutral-Spine 发表于 2025-3-28 14:04:00

http://reply.papertrans.cn/31/3078/307796/307796_40.png
页: 1 2 3 [4] 5 6
查看完整版本: Titlebook: Elliptic Integrals and Elliptic Functions; Takashi Takebe Textbook 2023 The Editor(s) (if applicable) and The Author(s), under exclusive l